How to determine phase shift of a sine waveform? [closed]

Question 1

As explained on ehow.com

Instructions:

1.) Measure the horizontal shift between two wave functions by graphing them. A shift to the right is a positive phase shift and a shift to the left is a negative phase shift.

2.) Determine the phase shift between the cosine function and the sine function. Use the trigonometry identity cos(x) = sin(x+Pi/2) to show that we can obtain the cosine function by shifting the sine wave Pi/2 to the left. The cosine function is therefore the sine function with a phase shift of -Pi/2.

3.) Generalize the sine wave function with the sinusoidal equation y = Asin(B[x - C]) + D. In this equation, the amplitude of the wave is A, the expansion factor is B, the phase shift is C and the amplitude shift is D.

4.) Express a wave function in the form y = Asin(B[x - C]) + D to determine its phase shift C. For example, for the function cos(x) = sin(x+Pi/2) = sin(x - [-Pi/2]), we have C = -Pi/2. Therefore, shifting the phase of the sine function by -Pi/2 will produce the cosine function.

5.) Calculate the phase shift of the function y = sin(2x - Pi/2). This function is equal to y = sin(2[x - Pi/4]) where A = 1, B = 2, C = Pi/4 and D = 0. The phase shift of y = sin(2x - Pi/2) is therefore Pi/4.

Question 2

You can see that the graph of the sine wave does not go through (0.0, 0.0). This fact alone tells you that there is a phase shift.

After telling that there is a phase shift, you want to know how much it is. To do this, you have to look where the point of the sine wave is which would normally go through (0.0, 0.0). Here, you can imagine that the wave would extend beyond the left border of the diagram until it reaches 0.0 on the y axis. That's called a zero-crossing. You now have to measure the distance on x between this zero-crossing and 0.0 on x. That's the phase shift.

Initially, from this diagram, your result is a phase shift measured in seconds, although this is valid, it is not the usual unit used for phase shifts. To get the phase shift in the usual unit, which is radian, use phi = timeDelta * f * 2pi.

Generally, the phase shift is the x-distance between (0.0, 0.0) and the next zero-crossing to the left, given in an angle, usually in radian, sometimes also given in degree.

Question 3

In high school mathematics, the assumption is that the phase shift is simply the same as the horizontal shift. This in many ways renders the concept of phase shift as something distinct from horizontal shift meaningless.

In signal analysis it's convenient to measure horizontal shift in both absolute terms, i.e. a shift of 3 milli seconds to the right, (in many applications time is the horizontal axis), but also in terms of the number of cycles you have shifted. One cycle is considered to be 360° or 2π radians.-

For example, suppose you have a signal v(t) = sin (100πt). The period is 0.02 s, or 20 mS. The frequency is 50 hz.

If we shift this signal to the right 0.002 s or 2 mS, the equation would be v(t) = sin (100π(t - 0.002))

The horizontal shift is exactly one tenth of a cycle. Phase shift simply assumes that one cycle is 360° (or 2π radians).

One tenth of this is 36°, or 0.2π radians.

Because the shift is to the right, or later in time, we can say the signal lags (or is behind) the unshifted signal one tenth of a cycle, or 36°.

Expanding what's inside the sin function: v(t) = sin (100πt - 0.2π). The "-" means it's lagging, and the 0.2π gives you the phase shift.

This is a sensible way to do things as the time axis is usually related to some rotating object. If you shift one full cycle, you have shifted 360°. Anyone who has ever skated boarded knows that a 'three sixty', bring you back to the same position. Shifting one full period, left or right, brings you back to the same signal.

To summarize: For a signal of period T, with a horizontal shift of h. The phase shift is h/T cycles (h/T) x 360° (h/T) x 2π radians.

A 360° phase shift is no shift at all.